Theory of Financial Risk and Derivative Pricing From Statistical Physics to Risk Management second edition

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Bouchaud, Jean-Philippe, 1962– Theory of financial risk and derivative pricing : from statistical physics to risk management / Jean-Philippe Bouchaud and Marc Potters.–2nd edn p. cm. Rev. edn of: Theory of financial risks. 2000. Includes bibliographical references and index. Contents Foreword page xiii Preface xv 1 Probability theory: basic notions 1 1.1 Introduction 1 1.2 Probability distributions 3 1.3 Typical values and deviations 4 1.4 Moments and characteristic function 6 1.5 Divergence of moments – asymptotic behaviour 7 1.6 Gaussian distribution 7 1.7 Log-normal distribution 8 1.8 Lévy distributions and Paretian tails 10 1.9 Other distributions (*) 1 4 1.10 Summary 16 2 Maximum and addition of random variables 17 2.1 Maximum of random variables 17 2.2 Sums of random variables 21 2.2.1 Convolutions 21 2.2.2 Additivity of cumulants and of tail amplitudes 22 2.2.3 Stable distributions and self-similarity 23 2.3 Central limit theorem 24 2.3.1 Convergence to a Gaussian 25 2.3.2 Convergence to a Lévy distribution 27 2.3.3 Large deviations 28 2.3.4 Steepest descent method and Cramèr function (*) 3 0 2.3.5 The CLT at work on simple cases 32 2.3.6 Truncated Lévy distributions 35 2.3.7 Conclusion: survival and vanishing of tails 36 2.4 From sum to max: progressive dominance of extremes (*) 3 7 2.5 Linear correlations and fractional Brownian motion 38 2.6 Summary 40 vi Contents 3 Continuous time limit, Ito calculus and path integrals 43 3.1 Divisibility and the continuous time limit 43 3.1.1 Divisibility 43 3.1.2 Infinite divisibility 44 3.1.3 Poisson jump processes 45 3.2 Functions of the Brownian motion and Ito calculus 47 3.2.1 Ito's lemma 47 3.2.2 Novikov's formula 49 3.2.3 Stratonovich's prescription 50 3.3 Other techniques 51 3.3.1 Path integrals 51 3.3.2 Girsanov's formula and the Martin–Siggia–Rose trick (